Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

No eleventh conditional Ingleton inequality

Tobias Boege

TL;DR

The paper constructs a rational binary distribution on four binary variables $(X,Y,Z,U)$ that satisfies the CI relations $[X \perp\!\perp Y]$, $[X \perp\!\!\perp Z | U]$, $[Y \perp\!\!\perp U | Z]$, and $[Z \perp\!\perp U | XY]$ while giving a negative Ingleton value $\square({XY|ZU}) \approx -0.00757$, resolving Open Question 1 and confirming that there are exactly ten inclusion-minimal CI sets (up to symmetry) that force Ingleton to hold for four discrete variables; it also settles the last remaining essential conditionality case for Ingleton-type CI inequalities. The authors combine algebraic-statistics methods, circuit/mask analysis, SAT-solving, and numerical optimization to identify the CI structures, derive a rational parametrization with a three-parameter semialgebraic model, and produce a concrete rational counterexample with $p_{1011}=\tfrac{2}{99}$, $p_{1111}=\tfrac{2}{11}$, $p_{0110}=\tfrac{10}{693}$ yielding $\square({XY|ZU})<0$ and an Ingleton violation of about $0.00757$. They further classify essential conditionality, showing that among a family of conditional Ingleton inequalities, only the one labeled (25cI) is essential, using both multiplier-based arguments and a curve of counterexamples parameterized by $\varepsilon \to 0$. Overall, the work fully characterizes when Ingleton-type inequalities are implied by CI assumptions on four discrete variables and provides a reproducible computational framework for verifying such phenomena.

Abstract

A rational probability distribution on four binary random variables $X, Y, Z, U$ is constructed which satisfies the conditional independence relations $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Y]$, $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Z \mid U]$, $[Y \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid Z]$ and $[Z \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid XY]$ but whose entropy vector violates the Ingleton inequality. This settles a recent question of Studený (IEEE Trans. Inf. Theory vol. 67, no. 11) and shows that there are, up to symmetry, precisely ten inclusion-minimal sets of conditional independence assumptions on four discrete random variables which make the Ingleton inequality hold. The last case in the classification of which of these inequalities are essentially conditional is also settled.

No eleventh conditional Ingleton inequality

TL;DR

The paper constructs a rational binary distribution on four binary variables that satisfies the CI relations , , , and while giving a negative Ingleton value , resolving Open Question 1 and confirming that there are exactly ten inclusion-minimal CI sets (up to symmetry) that force Ingleton to hold for four discrete variables; it also settles the last remaining essential conditionality case for Ingleton-type CI inequalities. The authors combine algebraic-statistics methods, circuit/mask analysis, SAT-solving, and numerical optimization to identify the CI structures, derive a rational parametrization with a three-parameter semialgebraic model, and produce a concrete rational counterexample with , , yielding and an Ingleton violation of about . They further classify essential conditionality, showing that among a family of conditional Ingleton inequalities, only the one labeled (25cI) is essential, using both multiplier-based arguments and a curve of counterexamples parameterized by . Overall, the work fully characterizes when Ingleton-type inequalities are implied by CI assumptions on four discrete variables and provides a reproducible computational framework for verifying such phenomena.

Abstract

A rational probability distribution on four binary random variables is constructed which satisfies the conditional independence relations \perp\mkern-10mu\perp, \perp\mkern-10mu\perp, \perp\mkern-10mu\perp and \perp\mkern-10mu\perp but whose entropy vector violates the Ingleton inequality. This settles a recent question of Studený (IEEE Trans. Inf. Theory vol. 67, no. 11) and shows that there are, up to symmetry, precisely ten inclusion-minimal sets of conditional independence assumptions on four discrete random variables which make the Ingleton inequality hold. The last case in the classification of which of these inequalities are essentially conditional is also settled.
Paper Structure (13 sections, 1 theorem, 24 equations, 1 figure)

This paper contains 13 sections, 1 theorem, 24 equations, 1 figure.

Table of Contents

  1. Summary
  2. On conditional Ingleton inequalities
  3. Ingleton inequality and entropy region
  4. Discrete representability of CI structures
  5. Masks and conditional Ingleton inequalities
  6. Construction of the distribution
  7. Circuits, masks and scores
  8. Parametrization of $\mathcal{L}_1$
  9. Numerical optimization and a rational point
  10. Classification of essentially conditional Ingleton inequalities
  11. Essential conditionality
  12. Sampling for a counterexample
  13. Remarks

Key Result

Theorem 1

On four discrete random variables $X, Y, Z, U$ there are precisely ten inclusion-minimal conditional independence assumptions which make Ingleton's inequality $\square ({XY|ZU})\ge 0$ hold for entropy vectors (up to the symmetries $X \leftrightarrow Y$ and $Z \leftrightarrow U$ in the Ingleton expre The conditional Ingleton inequalities given by eq:11cI--eq:15cI are not essentially conditional but

Figures (1)

  • Figure 1: The model $\mathcal{L}$ in its $(p_{1111}, p_{1011})$-parameter space $\mathcal{T}$. Points with a positive non-Ingleton score $\varrho_2$ are colored in red. The rational non-Ingleton distribution with $p_{1011} = 2/99$ and $p_{1111} = 2/11$ is marked with a black dot.

Theorems & Definitions (2)

  • Theorem
  • Remark 4.1